Practice Questions

Q71.Let f : R →R be defined as ⎧−55x, if x < −5 f(x) = −120x, if −5 ≤x ≤4 ⎨2x3 −3x2 ⎩2x3 −3x2 −36x −336, if x > 4 Let A = {x ∈R : f is increasing}. Then A is equal to: (1) (−5, ∞) (2) (−5, −4) ∪(4, ∞) (3) (−∞, −5) ∪(−4, ∞) (4) (−∞, −5) ∪(4, ∞)

Q71.Let 𝑓: 𝑅→𝑅 be defined as 𝜆𝑥2 - 5𝑥+ 6 𝑥< 2 𝜇5𝑥- 𝑥2 - 6 𝑓𝑥= tan ( 𝑥- 2 ) 𝑒 𝑥- [𝑥] 𝑥> 2 𝜇 𝑥= 2 where 𝑥 is the greatest integer less than or equal to 𝑥. If 𝑓 is continuous at 𝑥= 2, then 𝜆+ 𝜇 is equal to : (1) 𝑒( - 𝑒+ 1 ) (2) 𝑒( 𝑒- 2 ) (3) 1 (4) 2𝑒- 1

Q71.Let [x] denote the greatest integer ≤x, where x ∈R. If the domain of the real valued function f(x) = is (−∞, a) ∪[b, c) ∪[4, ∞), a < b < c, then the value of a + b + c is: √|[x]|−2|[x]|−3 (1) 8 (2) 1 (3) −2 (4) −3

Q72.If (sin−1 x)2 −(cos−1 x)2 = a; 0 < x < 1, a ≠0, then the value of 2x2 −1 is (1) cos( 2aπ ) (2) sin( 2aπ ) (3) cos( 4aπ ) (4) sin( 4aπ )

Q72.If the curve y = ax2 + bx + c, x ∈R, passes through the point (1, 2) and the tangent line to this curve at origin is y = x, then the possible values of a, b, c are: (1) a = −1, b = 1, c = 1 (2) a = 1, b = 0, c = 1 (3) a = 1, b = 1, c = 0 (4) a = 12 , b = 12 , c = 1

Q72.If P = [ ], 2 1 (1) [125 10 ] (2) [10 501 ] (3) [10 251 ] (4) [150 10 ] is equal to:

Q72.The triangle of maximum area that can be inscribed in a given circle of radius ' r' is : (1) An equilateral triangle having each of its side of (2) An isosceles triangle with base equal to 2r. length √3r. (3) An equilateral triangle of height 2r . (4) A right angle triangle having two of its sides of 3 length 2r and r. dt, then f(e) + f( 1e ) is equal to

Q72.If the following system of linear equations 2x + y + z = 5 x −y + z = 3 x + y + a z = b has no solution, then : (1) a = −13 , b ≠73 (2) a ≠13 , b = 73 (3) a ≠−13 , b = 73 (4) a = 13 , b ≠73

Q72.The function 𝑓𝑥= 𝑥3 - 6𝑥2 + 𝑎𝑥+ 𝑏 is such that 𝑓2 = 𝑓4 = 0. Consider two statements: 𝑆1 there exists 𝑥1, 𝑥2 ∈2, 4, 𝑥1 < 𝑥2, such that 𝑓'𝑥1 = - 1 and 𝑓'𝑥2 = 0 . 𝑆2 there exists 𝑥3, 𝑥4 ∈2, 4, 𝑥3 < 𝑥4, such that 𝑓 is decreasing in 2, 𝑥4, increasing in 𝑥4, 4 and 2𝑓'𝑥3 = √3𝑓𝑥4 then (1) 𝑆1 is true and 𝑆2 is false (2) both 𝑆1 and 𝑆2 are false (3) both 𝑆1 and 𝑆2 are true (4) 𝑆1 is false and 𝑆2 is true JEE Main 2021 (01 Sep Shift 2) JEE Main Previous Year Paper Q73. 𝜋 sec2𝑥𝑓(𝑥)d𝑥 4 ∫2 Let f : R →R be a continuous function. Then lim 𝜋2 is equal to: 𝑥→𝜋/ 4 𝑥2 - 16 (1) 𝑓( 2 ) (2) 2𝑓( √2 ) (3) 2𝑓( 2 ) (4) 4𝑓( 2 )

Q72.The number of solutions of the equation sin−1[x2 + 13 ] + cos−1[x2 −23 ] = x2 for x ∈[−1, 1], and [x] denotes the greatest integer less than or equal to x, is : (1) 2 (2) 0 (3) 4 (4) Infinite . Then f is:

Q72.Let A = [−11 24 ]. If A−1 = αI + βA, α, β ∈R, I is a 2 × 2 identity matrix, then 4(α −β) is equal to : (1) 5 (2) 83 (3) 2 (4) 4 (1 + |sin x|) |sin x| , −π4 < x < 0Q73. ⎧ 3a b , x = 0 Let f : (−π4 , π4 ) →R be defined as, f(x) = ⎨ ⎩ ecot 4x/ cot 2x , 0 < x < π4 If f is continuous at x = 0 then the value of 6a + b2 is equal to: (1) 1 −e (2) e −1 (3) 1 + e (4) e

Q72.The system of equations kx + y + z = 1, x + ky + z = k and x + y + zk = k2 has no solution if k is equal to: (1) 0 (2) 1 (3) −1 (4) −2

Q72.Let f be any function defined on R and let it satisfy the condition: |f(x) −f(y)| ≤(x −y)2 , ∀(x, y) ∈R. If f(0) = 1, then : (1) f(x) = 0, ∀x ∈R (2) f(x) can take any value in R (3) f(x) < 0, ∀x ∈R (4) f(x) > 0, ∀x ∈R

Q72.The number of elements in the set {x ∈R : (|x| −3)|x + 4| = 6} is equal to (1) 3 (2) 2 (3) 4 (4) 1

Q72.Let the system of linear equations 4x + λy + 2z = 0 2x −y + z = 0 μx + 2y + 3z = 0, λ, μ ∈R has a non-trivial solution. Then which of the following is true? JEE Main 2021 (18 Mar Shift 2) JEE Main Previous Year Paper (1) μ = 6, λ ∈R (2) λ = 2, μ ∈R (3) λ = 3, μ ∈R (4) μ = −6, λ ∈R

Q72.The sum of all the local minimum values of the twice differentiable function f : R →R defined by ′′(2) x + f ′′(1) is: f(x) = x3 −3x2 −3f 2 (1) −22 (2) 5 (3) −27 (4) 0

Q72.The function 𝑓𝑥= 4𝑥3 - 3𝑥2 - 2sin𝑥+ 2𝑥- 1cos𝑥: 6 1 1 (1) increases in 2, ∞ (2) decreases in - ∞, 2 1 1 (3) decreases in 2, ∞ (4) increases in - ∞, 2

Q72.The domain of the function cosec−1 ( 1+xx ) is : (1) [−12 , ∞) −{0} (2) (−1, −12 ] ∪(0, ∞) (3) [−12 , 0) ∪[1, ∞) (4) (−12 , ∞) −{0}

Q72.A function f(x) is given by f(x) = 5x+55x , then the sum of the series f( 201 ) + f( 202 ) + f( 203 ) + … + f( 2039 ) is equal to: (1) 19 (2) 49 2 2 (3) 39 (4) 29 2 2

Q72.If lim sin−1 x−tan−1 x is equal to L, then the value of (6L + 1) is x→0 3x3 (1) 1 (2) 1 6 2 (3) 6 (4) 2 JEE Main 2021 (18 Mar Shift 1) JEE Main Previous Year Paper Q73. 1 2 0 2 −1 5 Let A + 2B = ⎡ 6 −3 3⎤ and 2A −B = ⎡2 −1 6⎤ . If Tr(A) denotes the sum of all diagonal elements −5 3 1 0 1 2 ⎣ ⎦ ⎣ ⎦ of the matrix A, then Tr (A)−Tr (B) has value equal to (1) 1 (2) 2 (3) 0 (4) 3

Q72.Let 𝑓: [0, ∞) →[0, ∞) be defined as 𝑓𝑥= 𝑥𝑦𝑑𝑦 where [𝑥] is the greatest integer less than or equal to 𝑥. ∫0 Which of the following is true? (1) 𝑓 is continuous at every point in [0, ∞) and (2) 𝑓 is both continuous and differentiable except at differentiable except at the integer points. the integer points in [0, ∞) . (3) 𝑓 is continuous everywhere except at the integer (4) 𝑓 is differentiable at every point in [0, ∞) . points in [0, ∞) . 𝜋 𝜋

Q72.Let A and B be two 3 × 3 real matrices such that (A2 −B2) is invertible matrix. If A5 = B5 and A3 B2 = A2 B3, then the value of the determinant of the matrix A3 + B3 is equal to : (1) 2 (2) 4 (3) 1 (4) 0

Q72.A box open from top is made from a rectangular sheet of dimension a × b by cutting squares each of side x from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to: (1) a+b+√a2+b2−ab (2) a+b−√a2+b2−ab 6 12 (3) a+b−√a2+b2−ab (4) a+b−√a2+b2+ab 6 6

Q72.The domain of the function, 𝑓𝑥= sin-13𝑥2 + cos-1 2 ( 𝑥- 1 𝑥+ 1 ) 1 1 (1) 0, (2) 0, 2 4 (3) 1 1 ∪0 (4) -2, 0 ∪1 1 4, 2 4, 2

Q72.Let f, g : N →N such that f(n + 1) = f(n) + f(1) ∀ n ∈N and g be any arbitrary function. Which of the following statements is NOT true? (1) If f is onto, then f(n) = n∀n ∈N (2) If g is onto, then fog is one-one (3) f is one-one (4) If fog is one-one, then g is one-one